View source for Zero divisor

''in a ring or a semi-group with zero element''

A non-zero element such that the product with some non-zero element is zero. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z0992301.png" /> is called a left (right) divisor of zero if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z0992302.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z0992303.png" />) for at least one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z0992304.png" />.



====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z0992305.png" /> be a ring and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z0992306.png" /> a left module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z0992307.png" />. Then an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z0992308.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z0992309.png" /> is called a zero divisor in the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z09923010.png" /> if there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z09923011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z09923012.png" />.